August 2022

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Complex Analysis and Operator Theory

It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in ${\mathbb {C}}^{n},$ C n , apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in ${\mathbb {C}}^{n},n>1$ C n , n > 1 . From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion of k -fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family $S(k),k\ge 2,$ S ( k ) , k ≥ 2 , of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mapping f , from the Euclidean ball ${\mathbb {B}}^{n}$ B n into ${\mathbb {C}}^{n},$ C n , to belong to the family $S(k),k\ge 2.$ S ( k ) , k ≥ 2 . The result is obtained using an n -dimensional version of Jack’s Lemma.